Bake textures including light / shadows from camera's perspective
Here’s what the Blender documentation on Render Baking says about the Combined bake mode:

Bakes all materials, textures, and lighting except specularity.

Emphasis mine. The disparity you are seeing in your baked textures is the lack of specular reflections.
This should not be surprising, because specular reflections cannot be baked. This is because they are fundamentally dependent on viewing angle, even with completely static lighting. If you were to bake specularity from the perspective of the camera, it would look quite bad whenever the view changes.
The purpose of the Combined bake mode is to bake everything that can be baked in a viewpoint-independent way. It is not, however, a replacement for all further shading. What you should probably do instead is combine the baked texture with a glossy shader to provide viewpoint-dependent specular reflections. For example, you can bake a Combined pass including everything except the Glossy layer, then combine the result with a Glossy BSDF to add specular reflections on top:

You would want to create a similar setup in whatever game engine you are using (making sure to replicate the same lighting!). This is a totally standard workflow, and it’s still relatively cheap to compute.